
Results for the mass ratios
<FIGURE>The 1-\(\sigma \)  and 2-\(\sigma \)  HPD intervals of the ratios \(y_e/y_d\) , \(y_\mu /y_s\)  and \(y_\tau /y_b\)  at \(M_{Z}\)  for model 1 and model 2 with \(a = {\cal O}(1)\)  are shown in Figure REF . Since the posterior densities of these ratios do approximately not depend on the CSD variety, we show the HPD intervals only for one selected CSD variety for each model, namely \(\phi _{120}\)  for model 1 and \(\phi _{131}\)  for model 2. The 1-\(\sigma \)  and 2-\(\sigma \)  HPD intervals are indicated by dark and light colors, respectively, and are colored red (green) for model 1 (2). Moreover, indicated by dashed lines the current experimental central values are shown, while the grey areas represent the regions outside the experimental 1-\(\sigma \)  bounds. Note, that as discussed in Section REF , for the charged lepton Yukawa couplings we use a standard deviation of at least 1% to take into account theoretical uncertainties, e.g. from the limited precision of the RG running.
Figure REF  shows that model 1 predicts the ratio \(y_e/y_d\)  to be a bit smaller than the experimental 1-\(\sigma \)  range. On the other hand, the prediction by model 2 of this ratio spans from the experimental central value to the upper bound of the experimental 1-\(\sigma \)  range. The ratio \(y_\mu /y_s\)  is predicted to be within the 1-\(\sigma \)  range and around the upper bound of the 1-\(\sigma \)  range by the models 1 and 2, respectively. Both models predict a large range for the ratio \(y_\tau /y_b\)  which can (almost) be at the experimental central value.
In summary, our theoretical predictions of the ratios between charged lepton and down-type quark masses are well compatible with the experimental bounds.

Results for the neutrino sector
<FIGURE>Figure REF  visualizes the 1-\(\sigma \)  and 2-\(\sigma \)  HPD intervals of the neutrino observables \(\theta _{12}^{\text{PMNS}}\) , \(\theta _{13}^{\text{PMNS}}\) , \(\theta _{23}^{\text{PMNS}}\)  and \(\delta ^{\text{PMNS}}\)  at \(M_Z\) , represented by dark and light colors, respectively, for both CSD varieties of both models (with \(a = {\cal O}(1)\) ). The current experimental central values are indicated by dashed lines, whereas the areas outside the experimental 1-\(\sigma \)  range is represented by the grey areas. Since the \(\chi ^2\)  function of \(\theta _{23}^{\text{PMNS}}\)  has two minima, its plot contains two white regions. The global minimum of the experimental \(\chi ^2\)  function of \(\theta _{23}^{\text{PMNS}}\)  is indicated by a dashed line, while the local minimum is represented by a dotted line.
As it can be seen from Figure REF , the predicted range for \(\theta _{12}^{\text{PMNS}}\)  is much wider in model 1 than in model 2. The reason for this is that while in model 2 this observable is mainly fixed by the CSD3 structure of \(\textbf {Y}_\nu \) , in model 1 using CSD2 there is also a contribution to \(\theta _{12}^{\text{PMNS}}\)  from the charged lepton mixing matrix \(\textbf {E}_L\)  defined in Eq. (REF ). While for model 1 the predicted 1-\(\sigma \)  range of \(\theta _{12}^{\text{PMNS}}\)  has overlap with the current experimental 1-\(\sigma \)  range, for model 2 \(\theta _{12}^{\text{PMNS}}\)  is predicted to be slightly above the current experimental 1-\(\sigma \)  range. For both models the predicted 1-\(\sigma \)  range of \(\theta _{13}^{\text{PMNS}}\)  approximately coincides with the current experimental 1-\(\sigma \)  range. Concerning \(\theta _{23}^{\text{PMNS}}\)  in model 1 the predicted range depends on the choice of the CSD2 variety. For \(\phi _{102}\)  it overlaps with the second minimum of the experimental \(\chi ^2\)  distribution, while for \(\phi _{120}\)  it overlaps with the first minimum, but since the prediction in this case is not sharp it almost spans to the second minimum. In model 2, both CSD3 varieties can yield a prediction of \(\theta _{23}^{\text{PMNS}}\)  matching the local minimum (with lower value of \(\theta _{23}^{\text{PMNS}}\) ). Regarding \(\delta ^{\text{PMNS}}\) , while for model 1 the case \(\phi _{102}\)  yields a predicted 1-\(\sigma \)  range approximately overlapping the current experimental 1-\(\sigma \)  range, for \(\phi _{120}\)  the predicted 1-\(\sigma \)  range lies above the upper bound of the experimental 1-\(\sigma \)  range. Furthermore, both models predict \(\delta ^{\text{PMNS}}\)  to be at or above the current experimental 1-\(\sigma \)  range. In model 2, both cases of the CSD3 variety predict \(\delta ^{\text{PMNS}}\)  to be somewhat above the current experimental 1-\(\sigma \)  range. Note however that at higher \(\sigma \) -level the experimental range becomes very wide, so model 2 is not excluded by this discrepancy.

Results for the GUT scale and the masses of the added scalar fields
<FIGURE>Figure REF  shows the 1-\(\sigma \)  (dark) and 2-\(\sigma \)  (light) HPD intervals of the GUT scale \(M_{\text{GUT}}\)  and of the masses of the added scalar fields \(M_{\Phi _8}\) , \(M_{\Phi _6}\)  and \(M_{\Phi _{(3,3)}}\)  for model 1 (red) and 2 (green). For both models the cases \(a \ll 1\)  and \(a = {\cal O}(1)\)  are shown. While for model 1 we chose the CSD2 variety \(\phi _{120}\) , for model 2 the results for the CSD3 variety \(\phi _{131}\)  are shown.
From Figure REF  it can be clearly seen that the GUT scale \(M_{\text{GUT}}\)  is predicted to be significantly higher in the case \(a = {\cal O}(1)\)  compared to the case \(a \ll 1\)  for both models. This reinforces the results we got for the best-fit points (cf. Section REF ). The reason for this is — as discussed in Section REF  — that a higher GUT scale leads to better predictions for \(y_\tau /y_b\)  in the case \(a = {\cal O}(1)\) . On the other hand, there is no large difference in the prediction of the GUT scale between the two models if the same case is considered.
For the masses of the added scalar fields we get relatively similar predictions for both cases and for both models. Clearly, the masses of the extra particles can be above the reach of present and planned particle collider experiments, but discoveries could also be possible. Comparing the 2-\(\sigma \)  HPD results of these masses from the 2-loop calculation with the allowed ranges from the 1-loop estimation we did in Section REF , we notice that for \(M_{\Phi _{(3,3)}}\) , in the case of the 2-loop calculation, larger values are allowed than for the 1-loop estimation.

Results for the neutrino parameters
The 1-\(\sigma \)  (dark) and 2-\(\sigma \)  (light) HPD intervals of the neutrino coupling \(a\)  and the right-handed neutrino mass \(M_A\)  at the GUT scale \(M_{\text{GUT}}\)  for model 1 (red) and model 2 (green) for the case \(a = {\cal O}(1)\)  and the CSD variety \(\phi _{120}\)  (model 1), respectively \(\phi _{131}\)  (model 2) are shown in Figure REF .
In both models the neutrino coupling \(a\)  is bounded from the top by 2.4 since we set this bound by hand, as discussed in Section REF . The lower bound is in model 1 predicted to be smaller than in model 2. At 1-\(\sigma \)  (2-\(\sigma \) ) it is given by 2.08 (1.60) in model 1, respectively 2.22 (1.93) in model 2.
The prediction right-handed neutrino mass \(M_A\)  is similar in both models, but in model 1 the upper limit of the 1-\(\sigma \)  and 2-\(\sigma \)  regions are larger than in model 2.

Results for the nucleon decay rates
We now compute the HPD intervals of the 13 different nucleon decay channels listed in Table REF . These predictions are approximately independent of the choice of the CSD2 or CSD3 variety. We therefore choose the variety \(\phi _{120}\)  (\(\phi _{131}\) ) for model 1 (2). The main contribution to the theoretical uncertainty comes from the GUT scale \(M_{\text{GUT}}\)  since the decay rates are proportional to the inverse of the GUT scale to the fourth power \(\Gamma \propto (M_{\text{GUT}})^{-4}\) . However, the ratios of specific decay channels are independent on the GUT scale \(M_{\text{GUT}}\) . We will therefore also analyze these quantities to discriminate between the two models.
Figure REF  and REF  show the 1-\(\sigma \)  (dark) and 2-\(\sigma \)  (light) HPD intervals of the various decay channels and selected ratios of specific decay channels, respectively. For model 1 (2) the HPD intervals are colored red (green). In Figure REF  the current experimental bounds are indicated by blue lines.
Figure REF  shows that none of the two models is excluded by the current experimental bounds on the decay rates of the different nucleon decay channels.
In both models the dominant proton decay channel is \(p\rightarrow \pi ^0e^+\) , while the dominant neutron decay channel is \(n\rightarrow \pi ^-e^+\) . With the future sensitivity of \(2.0\cdot 10^{-67}\,\,\mathrm {GeV}\)  for the lifetime of the decay channel \(p\rightarrow \pi ^0e^+\)  at HyperKamiokande after 15 years of runtime [1]}, part of the predicted 2-\(\sigma \)  region will be tested. Further channels close to the experimental bound are \(p\rightarrow K^0\mu ^+\) , \(p\rightarrow K^+\bar{\nu }\)  and \(p\rightarrow \pi ^0\mu ^+\) . The decays channels with a pion in the final state as well as the ones with a kaon always dominate that of the eta meson. Figure REF  further indicates that for a given decay channel the 2-\(\sigma \)  HPD intervals always overlap. Also, the predicted ranges are not sharp — the 2-\(\sigma \)  HPD range spans over 3 to 4 orders of magnitude for each decay channel. The reason for this is the dependence of the decay rates on the inverse of the GUT scale to the fourth power. Therefore, with a measurement of a single decay channel alone it is not possible to discriminate between the two models. Instead, the ratios between different decay channels are needed.
To draw Figure REF  we first computed for each point in the MCMC dataset the ratio and only then determined the HPD intervals. There are more than 30 ratios with non-overlapping 2-\(\sigma \)  HPD intervals. On the other hand, there are several decay widths for which an exact ratio is predicted because of the form of the low-energy effective Lagrangian. The following ratios have in every SU(5) GUT the exact value \(1/2\) :
\(\frac{\Gamma _{p\rightarrow \pi ^0e^+}}{\Gamma _{n\rightarrow \pi ^-e^+}},\hspace{28.45274pt}\frac{\Gamma _{p\rightarrow \pi ^0\mu ^+}}{\Gamma _{n\rightarrow \pi ^-\mu ^+}},\hspace{28.45274pt}\frac{\Gamma _{n\rightarrow \pi ^0\bar{\nu }}}{\Gamma _{p\rightarrow \pi ^+\bar{\nu }}}.\) 
Moreover, since in this paper we only consider gauge boson mediated nucleon decay and neglect any contributions to the decay widths from the scalar fields, the following ratios are further fixed in our two models:
\(\frac{\Gamma _{p\rightarrow \pi ^0e^+}}{\Gamma _{p\rightarrow \eta ^0e^+}},\hspace{28.45274pt}\frac{\Gamma _{p\rightarrow \pi ^0\mu ^+}}{\Gamma _{p\rightarrow \eta ^0\mu ^+}},\hspace{28.45274pt}\frac{\Gamma _{p\rightarrow \pi ^+\bar{\nu }}}{\Gamma _{n\rightarrow \eta ^0\bar{\nu }}}.\) 
Their exact value depends on the coupling constants for the interaction between baryons and mesons \(F\approx 0.463\)  and \(D\approx 0.804\)  (cf. [2]}, [3]}) and is given by:
\(\frac{(1+F+D)^2}{3(-\frac{1}{3}+F-\frac{1}{3}D)^2}\approx 197.66\,.\) 
<FIGURE>Finally, by combining ratios from Eq. (REF ) with the ratios listed in Eq. (REF ) further fixed ratios can be constructed.
In Figure REF  we picked for each fixed ratio defined in Eq. (REF ) and (REF ) one representative giving us nine independent ratios with non-overlapping 2-\(\sigma \)  HPD intervals. All other ratios with non-overlapping 2-\(\sigma \)  HPD intervals can then be reconstructed from the ones given in Figure REF  using Eq. (REF ) and (REF ). Measuring one such ratio already allows to discriminate between the two models. On the other hand, by experimentally determining multiple ratios a model can be excluded.
The decay channels which are predicted to be close to the experimental bounds deliver candidates for the most interesting ratios. Furthermore, upcoming experiments are more probable to measure ratios close to \(1:1\)  than ratios in which the enumerator and denominator differ by several orders of magnitude. Therefore, the most interesting ratios are the ones where one of the three decay channels \(p\rightarrow \pi ^0\mu ^+\) , \(n\rightarrow \pi ^-\mu ^+\)  and \(p\rightarrow K^0e^+\)  is compared to a decay channel with a pion in the final state.
Note that despite the different construction of the neutrino sector in the two models, all ratios between two decay channels both having neutrinos in the final states have overlapping 2-\(\sigma \)  intervals, and therefore do not allow to discriminate between the different models. The reason for this is that if neutrinos are in the final state, we sum over the neutrino flavor states, and thus the flavor information is lost.

